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We construct measures of uncertainty and its dispersion exploiting the heterogeneity of a large set of model predictions. The approach is forward-looking, can be computed in real-time, and can be applied at any frequency. We illustrate the methodology with expected shortfall predictions of worldwide equity indices generated from 71 risk models. We use the new measures in asset pricing, risk forecasting, and for explaining the aggregate trading volume of S&P 500 firms.

Change-point processes are one flexible approach to model long time series. We propose a method to uncover which model parameter truly vary when a change-point is detected. Given a set of breakpoints, we use a penalized likelihood approach to select the best set of parameters that changes over time and we prove that the penalty function leads to a consistent selection of the true model. Estimation is carried out via the deterministic annealing expectation-maximization algorithm. Our method accounts for model selection uncertainty and associates a probability to all the possible time-varying parameter specifications. Monte Carlo simulations highlight that the method works well for many time series models including heteroskedastic processes. For a sample of 14 Hedge funds (HF) strategies, using an asset based style pricing model, we shed light on the promising ability of our method to detect the time-varying dynamics of risk exposures as well as to forecast HF returns.

Change-point (CP) VAR models face a dimensionality curse due to the proliferation of parameters that arises when new breaks are detected. To handle large data sets, we introduce the Sparse CP-VAR model that determines which parameters truly vary when a break is detected. By doing so, the number of new parameters to estimate at each regime is drastically reduced and the CP dynamic becomes easier to interpret. The Sparse CP-VAR model disentangles the dynamics of the mean parameters and the covariance matrix. The former uses CP dynamics with shrinkage prior distributions while the latter is driven by an infinite hidden Markov framework. A simulation study highlights that the framework detects correctly the number of breaks per model parameter, and that it takes advantage of common breaks in the cross-sectional dimension to more precisely estimate them. Our applications on financial and macroeconomic systems highlight that the Sparse CP-VAR model helps interpreting the detected breaks. It turns out that many spillover effects have zero regimes meaning that they are zero for the entire sample period. Forecasting wise, the Sparse CP-VAR model is competitive against several recent time-varying parameter and CP-VAR models in terms of log predictive densities.

Change-point time series specifications constitute flexible models that capture unknown structural changes by allowing for switches in the model parameters. Nevertheless most models suffer from an over-parametrization issue since typically only one latent state variable drives the switches in all parameters. This implies that all parameters have to change when a break happens. To gauge whether and where there are structural breaks in realized variance, we introduce the sparse change-point HAR model. The approach controls for model parsimony by limiting the number of parameters which evolve from one regime to another. Sparsity is achieved thanks to employing a nonstandard shrinkage prior distribution. We derive a Gibbs sampler for inferring the parameters of this process. Simulation studies illustrate the excellent performance of the sampler. Relying on this new framework, we study the stability of the HAR model using realized variance series of several major international indices between January 2000 and August 2015.

A new model - the factorial hidden Markov volatility (FHMV) model - is proposed for financial returns and their latent variances. It is also applicable to model directly realized variances. Volatility is modeled as a product of three components: a Markov chain driving volatility persistence, an independent discrete process capable of generating jumps in the volatility, and a predictable (data-driven) process capturing the leverage effect. An economic interpretation is attached to each one of these components. Moreover, the Markov chain and jump components allow volatility to switch abruptly between thousands of states, and the transition matrix of the model is structured in such a way as to generate a high degree of volatility persistence. In-sample results on six financial time series highlight that the FHMV process compares favorably to state-of-the-art volatility models. A forecasting experiment shows that it also outperforms its competitors when predicting volatility over time horizons ranging from one to one hundred days.

Markov-switching models are usually specified under the assumption that all the parameters change when a regime switch occurs. Relaxing this hypothesis and being able to detect which parameters evolve over time is relevant for interpreting the changes in the dynamics of the series, for specifying models parsimoniously, and may be helpful in forecasting. We propose the class of sticky infinite hidden Markov-switching autoregressive moving average models, in which we disentangle the break dynamics of the mean and the variance parameters. In this class, the number of regimes is possibly infinite and is determined when estimating the model, thus avoiding the need to set this number by a model choice criterion. We develop a new Markov chain Monte Carlo estimation method that solves the path dependence issue due to the moving average component. Empirical results on macroeconomic series illustrate that the proposed class of models dominates the model with fixed parameters in terms of point and density forecasts. Appendix available at: https://ssrn.com/abstract=2965668

Sequential Monte Carlo (SMC) methods are widely used for non-linear filtering purposes. Nevertheless the SMC scope encompasses wider applications such as estimating static model parameters so much that it is becoming a serious alternative to Markov-Chain Monte-Carlo (MCMC) methods. Not only SMC algorithms draw posterior distributions of static or dynamic parameters but additionally provide an estimate of the marginal likelihood. The tempered and time (TNT) algorithm, developed in the paper, combines (off-line) tempered SMC inference with on-line SMC inference for drawing realizations from many sequential posterior distributions without experiencing a particle degeneracy problem. Furthermore, it introduces a new MCMC rejuvenation step that is generic, automated and well-suited for multi-modal distributions. As this update relies on the wide heuristic optimization literature, numerous extensions are already available. The algorithm is notably appropriate for estimating Change-point models. As an example, we compare Change-point GARCH models through their marginal likelihoods over time.

This Appendix contains additional empirical results with respect to the published article. In Section 1, the posterior results for the HDP parameters of the IHMS- ARMA models are presented for the U.S. GDP growth rate and inflation series. In Section 2, we report additional in-sample and forecasting results for the same series. In Section 3, some results for a different truncation choice of the number of regimes in the approximate model are reported. Full paper available at: https://ssrn.com/abstract=2965441

We present an estimation and forecasting method, based on a differential evolution MCMC method, for inference in GARCH models subjected to an unknown number of structural breaks at unknown dates. We treat break dates as parameters and determine the number of breaks by computing the marginal likelihoods of competing models. We allow for both recurrent and non-recurrent (change-point) regime specifications. We illustrate the estimation method through simulations and apply it to seven financial time series of daily returns. We find structural breaks in the volatility dynamics of all series and recurrent regimes in nearly all series. Finally, we carry out a forecasting exercise to evaluate the usefulness of structural break models.

GARCH volatility models with fixed parameters are too restrictive for long time series due to breaks in the volatility process. Flexible alternatives are Markov-switching GARCH and change-point GARCH models. They require estimation by MCMC methods due to the path dependence problem. An unsolved issue is the computation of their marginal likelihood, which is essential for determining the number of regimes or change-points. We solve the problem by using particle MCMC, a technique proposed by Andrieu, Doucet, and Holenstein (2010). We examine the performance of this new method on simulated data, and we illustrate its use on several return series.